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Master's Dissertation
DOI
10.11606/D.45.2008.tde-06072012-140351
Document
Author
Full name
Renato Belinelo Bortolatto
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2008
Supervisor
Committee
Tal, Fabio Armando (President)
Buzzi, Claudio Aguinaldo
Garcia, Manuel Valentim de Pera
Title in Portuguese
Energia cinética e pontos de equilíbrio de sistemas hamiltonianos
Keywords in Portuguese
bacia de atração
energia cinética
Sistemas Hamiltonianos
Abstract in Portuguese
Estudaremos uma influência não trivial da energia cinética sobre pontos de equilébrio de sistemas Hamiltonianos a partir da segunda parte do artigo de Garcia & Tal "The influence of the kinetic energy in equilibrium of Hamiltonian systems". Nesse artigo demonstra-se, para um exemplo explícito de Hamiltonianos C(R4) definidos por Hi = Ti + para i {1,2}, que as bacias de atração de H1 e H2 são subvariedades de R4 com dimensão distinta. Discutiremos aqui de que forma esse resultado está relacionado com o estudo da estabilidade segundo Liapunov de pontos de equilíbrio de sistemas Hamiltonianos, em especial com a busca de uma inversão para o celebrado teorema de Dirichlet-Lagrange. Por fim apresentamos um novo teorema que estende o resultado acima para toda uma família de energias potenciais ,,k. A saber, mostramos que, se os parâmetros ,,k satisfazem a um simples critério aritmético então as bacias de atração de Hi = Ti + ,,k tem dimensões distintas para i {1, 2}.
Title in English
Kinetic energy and equilibrium points of Hamiltonian systems
Keywords in English
attraction basin
Hamiltonian systems
kinetic energy
Abstract in English
We study a non trivial influence of the kinetic energy on equilibrium points of Hamiltonian systems following the second part of Garcia & Tal article "The influence of the kinetic energy in equilibrium of Hamiltonian systems". In this article the authors show, for an explicit example of C (R4 ) Hamiltonians defined by Hi = Ti + for i {1, 2}, that the attraction basins of H1 and H2 have distinct dimensions as submanifolds of R4. Well discuss how this result is related to the study of the stability according to Liapunov of equilibrium points of Hamiltonian systems and especially how it is related to the inversion of the celebrated Lagrange-Dirichlet theorem. Finally well prove a new theorem which extends the result above for a whole family of potential energies ,,k. We show that, if the parameters ,,k satisfy a simple arithmetical criteria then the attraction basins of Hi = Ti + ,,k have different dimensions for i {1, 2}.
 
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ECPESH.pdf (333.06 Kbytes)
Publishing Date
2012-07-11
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • Bortolatto, R. B., GARCIA, M. V. P., and TAL, F. A. Kinetic Energy and the Stable Set [doi:10.1007/s12346-010-0029-2]. Qualitative Theory of Dynamical Systems [online], 2011, vol. 10, p. 1-10.
All rights of the thesis/dissertation are from the authors
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